Creatures in Flight · Quiz 32 · Scientific Notation

The Guide

Before you begin

This quiz is about a way of writing numbers that are very large or very small: scientific notation, in which a number is written as a number between 1 and 10, multiplied by a power of ten. Three thousandths of a kilogram becomes 3 × 10⁻³ kg. A billion wingbeats becomes 1 × 10⁹. The notation is not a new kind of number, only a new way of writing the same numbers, and it exists because ordinary writing runs out of room at both ends. If you did Quiz 15, on powers of ten, and Quiz 31, on negative exponents, you have the two pieces already; if not, everything you need is below. The hummingbird at the top of the page is the reason the notation is worth learning: one animal whose numbers run from hundred-thousandths to billions.

Powers of ten, and what the exponent counts

A power of ten is a 1 followed by zeros, and the exponent counts the zeros: 10¹ = 10, 10² = 100, 10³ = 1,000, 10⁶ = 1,000,000. A better way to say the same thing, because it will still be true below zero: the exponent counts how many places the 1 sits from the units place. In 1,000 the 1 is three places to the left of the units, so the exponent is 3. Now the other direction. 10⁻¹ is one tenth, 0.1, and 10⁻² is one hundredth, 0.01, and 10⁻³ is one thousandth, 0.001, because a negative exponent puts the factors on the bottom of a fraction. In 0.001 the 1 sits three places to the right of the units place, past the decimal point, and the exponent is −3. Same rule, same count, the sign telling you which way.

Open the ladder of powers of ten, from a million to a millionth

Power	Written out	Its name
10⁶	1,000,000	a million
10⁵	100,000	a hundred thousand
10⁴	10,000	ten thousand
10³	1,000	a thousand
10²	100	a hundred
10¹	10	ten
10⁰	1	one
10⁻¹	0.1	a tenth
10⁻²	0.01	a hundredth
10⁻³	0.001	a thousandth
10⁻⁴	0.0001	a ten-thousandth
10⁻⁵	0.00001	a hundred-thousandth
10⁻⁶	0.000001	a millionth

The chart is wider than your screen: slide it sideways with your finger, or turn your phone.

Writing a big number: the point moves left

Take 5,300, whose digits are 5, 3, 0, 0. Every whole number ends in a decimal point you cannot see, so 5,300 is 5,300. with the point after the last zero. Move the point to just after the first digit that is not zero, the 5, and it becomes 5.3, which is a number between 1 and 10. Moving the point 3 places to the left divided the number by 10 three times, by 10³, so to keep the value the same you multiply back by 10³: 5,300 = 5.3 × 10³. Check it by reading it back: 5.3 × 1,000 = 5,300. The same move writes 0.003, whose digits after the point are 0, 0, 3. Its point is three places to the left of the 3, so move it 3 places to the right, to just after the 3, and the number becomes 3. That multiplied by 10 three times, so multiply by 10⁻³ to undo it: 0.003 = 3 × 10⁻³.

The point moves to just after the first digit that is not zero. Count the places it moved: left for a big number, and the exponent is that count; right for a small one, and the exponent is that count with a minus sign.

So the rule, once the picture has shown it: move the point to just after the first nonzero digit, count the places, and that count is the exponent, positive if the point moved left and negative if it moved right. Here it is on the bird. A ruby-throated hummingbird’s heart beats about 1,200 times a minute in flight, which is 1.2 × 10³ beats a minute. It has about 940 feathers, few for a bird, and 940 is 9.4 × 10². It beats its wings about 53 times a second, which is 5.3 × 10¹. And it weighs about 3 grams, which is 0.003 kilograms, which is 3 × 10⁻³ kg.

Why between 1 and 10? Every number can be written many ways with powers of ten: 5,300 is also 53 × 10² and 0.53 × 10⁴. Scientific notation picks one of them so that two people writing the same number write the same thing, and the one it picks keeps the front number between 1 and 10, one digit before the point. That choice has a payoff: the exponent alone tells you the size of the number, because a front number between 1 and 10 can only nudge it, never carry it into the next power. When a calculation leaves you with a front number outside that range, 53 × 10² say, move the point once more and adjust the exponent by one: 5.3 × 10³.

Reading it back

To read 8.64 × 10⁴, move the point 4 places to the right, filling with zeros where the digits run out: 8.64 becomes 86,400. That is the number of seconds in a day, 24 × 60 × 60, and the check is that multiplying it out gives 86,400 too. To read 1.9 × 10⁻², move the point 2 places to the left, filling with zeros: 0.019. That is about how long one wingbeat takes, one second divided by 53, a little under two hundredths of a second.

Which is bigger: read the exponent first

Because the front number is always between 1 and 10, the exponent settles size on its own. 1 × 10⁴ is bigger than 9 × 10³, even though 9 is bigger than 1, because 10,000 is bigger than 9,000; the exponent 4 beats the exponent 3 before the front numbers are looked at. Only when two exponents are the same do the front numbers decide: 9 × 10³ is bigger than 2 × 10³. The same holds below zero, where it catches more people. 3 × 10⁻² is bigger than 3 × 10⁻³, because three hundredths is bigger than three thousandths, and −2 is bigger than −3, as Quiz 20 showed on the number line. Compare exponents first. If they tie, compare front numbers.

Multiplying and dividing: Quiz 31’s rules do the work

Multiply 3 × 10³ by 2 × 10⁴. The front numbers multiply: 3 × 2 = 6. The powers of ten multiply, and powers with the same base are multiplied by adding the exponents, which Quiz 31 called ADD: 10³ × 10⁴ = 10⁷. So the product is 6 × 10⁷. Check with the numbers written out: 3,000 × 20,000 = 60,000,000, which is 6 followed by seven zeros. Dividing works the same way with SUBTRACT: 8 × 10⁵ divided by 4 × 10² is 8 ÷ 4 = 2 in front and 10^{5 − 2} = 10³ behind, so 2 × 10³, and 800,000 ÷ 400 = 2,000 agrees.

Sometimes the front numbers step outside 1 to 10. Multiply 5 × 10³ by 4 × 10²: 5 × 4 = 20 and 10³ × 10² = 10⁵, so 20 × 10⁵. That is a true number, but not yet in the notation, because 20 is not between 1 and 10. Move the point one place left, which divides 20 by 10, and raise the exponent by one to pay it back: 2 × 10⁶. Check: 5,000 × 400 = 2,000,000.

Adding is different, and Marcus has the wrong tool for it. To add 3 × 10⁴ and 2 × 10³, Marcus adds the front numbers and adds the exponents: 5 × 10⁷. He has picked up ADD, the rule for multiplying powers, and used it on an addition. It is his habit, the right tool in the wrong room, the way he flipped the wrong fraction in Quiz 12, divided the wrong way around in Quiz 16, used multiplying’s sign rule on an addition in Quiz 22, set up a proportion where the relationship was additive in Quiz 25, added the sides of a room and called it an area in Quiz 27, and halved to find a square root in Quiz 30. The check is the numbers themselves: 30,000 + 2,000 = 32,000, which is 3.2 × 10⁴, and his answer is fifty million. Adding and subtracting need the same exponent, so that the front numbers count the same thing, the way adding fractions in Quiz 10 needed the same denominator. Rewrite the smaller one to match the larger: 2 × 10³ is 0.2 × 10⁴, and now 3 + 0.2 = 3.2 with the 10⁴ kept: 3.2 × 10⁴.

The bird, from one end of the line to the other

Here is what the notation is for. Below is one line of powers of ten, from 10⁻⁵ up to 10⁹, one mark for each exponent, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, so that each mark is ten times the one before it. On it are the numbers of one ruby-throated hummingbird. The grip of a single hooklet, one of the tiny hooks that zip a feather’s barbs together, was measured at about 1.4 × 10⁻⁵ newtons. The bird weighs 3 × 10⁻³ kg. One wingbeat takes about 1.9 × 10⁻² seconds, and there are 5.3 × 10¹ of them in a second. Its heart beats 1.2 × 10³ times a minute. A day has 8.64 × 10⁴ seconds. And a model of its lifetime, worked in question 8, comes to about 1.1 × 10⁹ wingbeats. The same bird, one line, fourteen powers of ten apart at its ends.

One animal, written on one line. Without the notation the two ends would be 0.000014 and 1,100,000,000, and the zeros would have to be counted by hand every time.

A calculator’s short form. When a calculator runs out of room, it shows a number in scientific notation in a short form, usually 8.64 × 10⁴ written as 8.64E4 or 8.64×10⁴, depending on the model. The E means “times ten to the.” If you see 1.1E9 on the test’s calculator, read it as 1.1 × 10⁹, a number with ten digits before the point.

Worked Examples

Three to study before you start

Example 1 · Writing the notation, both directions

The hummingbird makes its Gulf of Mexico crossing, about 800 kilometers, without landing. Write 800 in scientific notation: the invisible point is after the last zero, 800., and it moves 2 places left to sit after the 8, so 800 = 8 × 10² km. Check: 8 × 100 = 800. Now a barb of a feather. A study of feathers under an electron microscope measured the segments of a barb at 60 micrometers, which is 0.00006 meters. The point moves 5 places right to sit after the 6, so 0.00006 = 6 × 10⁻⁵ m. Check: 6 × 0.00001 = 0.00006. Left for large with a positive exponent, right for small with a negative one, and the count of places is the exponent either way.

Example 2 · Reading it back, and comparing

Cornell reports the oldest known ruby-throated hummingbird at a little over 9 years. Suppose a bird lived 3.3 × 10³ days. Read it back: move the point 3 places right, 3.3 becomes 3,300, so 3,300 days, about nine years. Now compare: is 3.3 × 10³ days more or less than 1 × 10⁴ days? Exponents first: 4 beats 3, so 1 × 10⁴ days, which is 10,000 days, is the larger, even though 3.3 is bigger than 1. Then the small end: is 1.9 × 10⁻² seconds, one wingbeat, more or less than 5 × 10⁻³ seconds? Exponents first: −2 is bigger than −3, so 1.9 × 10⁻² is larger; 0.019 against 0.005 agrees.

Example 3 · Multiplying, and putting the front number back in range

How many wingbeats in a minute? 53 a second, 60 seconds: 5.3 × 10¹ × 6 × 10¹. Front numbers: 5.3 × 6 = 31.8. Powers of ten, by ADD: 10¹ × 10¹ = 10². So 31.8 × 10², and 31.8 is not between 1 and 10. Move the point one place left and raise the exponent by one: 3.18 × 10³, which is 3,180 wingbeats a minute. Check the long way: 53 × 60 = 3,180. The two steps in the notation did exactly what the one multiplication did, and they will keep doing it when the numbers are too large to multiply the long way.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. After every answer in scientific notation, read it back once as an ordinary number to see that it is the size you meant.

Write 1,200 heartbeats a minute and 53 wingbeats a second in scientific notation.
Write 0.003 kg, the bird’s mass, and 0.000014 N, one hooklet’s grip, in scientific notation.
Marcus added 3 × 10⁴ and 2 × 10³ and wrote 5 × 10⁷. Which statement names what he did?
- A) He added the exponents, which is the rule for multiplying powers, not for adding. Adding needs the same exponent: 30 × 10³ + 2 × 10³ = 32 × 10³ = 3.2 × 10⁴.
- B) Nothing: 5 × 10⁷ is correct, because when powers of ten are added their exponents add.
- C) He should have multiplied the exponents, getting 5 × 10¹².
- D) He was right to add the exponents but should have multiplied the front numbers, getting 6 × 10⁷.
Write 8.64 × 10⁴ seconds, one day, and 1.9 × 10⁻² seconds, one wingbeat, as ordinary numbers.
Which is larger in each pair, and how do you know without writing the numbers out? First 9 × 10³ or 1 × 10⁴. Then 3 × 10⁻³ or 3 × 10⁻².
There are 3.18 × 10³ wingbeats in a minute. Multiply by 6 × 10¹ minutes to find the wingbeats in an hour, and write the answer in scientific notation.

A ruby-throated hummingbird hanging in the air beneath a spike of pink flowers, wings blurred, bill in a blossom

ReadingA reading rest stop, for the stubborn ones.

Read the Ruby-throated Hummingbird page at All About Birds, from the Cornell Lab of Ornithology. Land on the list headed Cool Facts, near the foot of the page: the 53 wingbeats a second are there, and so is the oldest bird on record, a female of at least 9 years and 2 months, caught and released in West Virginia in 2014.

The photograph above is Jim Hudgins’s, for the U.S. Fish and Wildlife Service; it is not from Cornell’s page.

The bird crosses the Gulf of Mexico, about 8 × 10² km, in about 2 × 10¹ hours without landing. Divide to find its speed in kilometers per hour, in scientific notation and as an ordinary number.
A simplified model of the bird’s lifetime: 53 wingbeats a second, 4 hours of flying a day, 365 days a year, for 4 years. Which is closest to the number of wingbeats in that life?
- A) 1 × 10⁶
- B) 1 × 10⁷
- C) 1 × 10⁸
- D) 1 × 10⁹
Quiz 29’s locusts: the Food and Agriculture Organization counts about 8 × 10⁷ locusts in each square kilometer of a dense swarm. How many locusts are in a swarm that covers 5 × 10² square kilometers? Answer in scientific notation.
One hooklet holds with a force of about 1.4 × 10⁻⁵ N, and in the feather study about 20 of them let go at once. Multiply to find the force that takes, in scientific notation. The study’s measured average was 2.7 × 10⁻⁴ N; say how close the multiplication comes.

Answer Key

Check your work

Open the key: after you've finished all ten

1 1.2 × 10³ and 5.3 × 10¹

In 1,200 the point moves 3 places left to sit after the 1, so the exponent is 3. In 53 it moves 1 place, so the exponent is 1. Read each back to check: 1.2 × 1,000 = 1,200 and 5.3 × 10 = 53. If you wrote 12 × 10², the value is right but the front number is not between 1 and 10; move the point once more.

2 3 × 10⁻³ kg and 1.4 × 10⁻⁵ N

In 0.003 the point moves 3 places right to sit after the 3; in 0.000014 it moves 5 places right to sit after the 1. Right means a negative exponent. Read back: 3 × 0.001 = 0.003, and 1.4 × 0.00001 = 0.000014. If you wrote 14 × 10⁻⁶ for the second, the value is right and the front number is out of range: move the point one place left and raise the exponent from −6 to −5.

3 A.

Adding needs the same exponent, so that the front numbers count the same thing; then the front numbers add and the exponent is kept: 30,000 + 2,000 = 32,000 = 3.2 × 10⁴. Marcus used ADD, the rule for multiplying powers, on an addition: the right tool in the wrong room, as in Quizzes 12, 16, 22, 25, 27 and 30. B) states his error as the rule, and 50,000,000 for a sum of two numbers under 40,000 refuses it. C) is the multiplying error made worse. D) would give 6 × 10⁷, which is what 3 × 10⁴ × 2 × 10³ equals, a product and not a sum.

4 86,400 seconds and 0.019 seconds

A positive exponent moves the point right, 4 places, filling with zeros: 8.64 becomes 86,400, which is 24 × 60 × 60. A negative exponent moves it left, 2 places, filling with zeros: 1.9 becomes 0.019. If you wrote 0.0019 for the second, the point moved one place too many; the exponent counts the places, and it says 2.

5 1 × 10⁴, and 3 × 10⁻²

Exponents first. 4 beats 3, so 1 × 10⁴ is larger, 10,000 against 9,000, though 9 is bigger than 1. And −2 is bigger than −3, so 3 × 10⁻² is larger, three hundredths against three thousandths. The front numbers only decide when the exponents tie, and here they do not. If you chose 9 × 10³, the front number was read as the size, which it never is in this notation.

6 1.908 × 10⁵, about 190,800 wingbeats an hour

Front numbers: 3.18 × 6 = 19.08. Exponents, by ADD: 3 + 1 = 4, so 19.08 × 10⁴. Then 19.08 is not between 1 and 10: move the point one place left and raise the exponent to 5, 1.908 × 10⁵. Check the long way: 3,180 × 60 = 190,800. If you left it as 19.08 × 10⁴, the value is right and the writing is not finished.

7 4 × 10¹ km per hour, which is 40 km per hour

Front numbers: 8 ÷ 2 = 4. Exponents, by SUBTRACT: 2 − 1 = 1. So 4 × 10¹, and reading it back gives 40. Check: 800 ÷ 20 = 40. A bird weighing three grams, holding 40 kilometers an hour for twenty hours over open water, is the reason the crossing gets written about.

8 D.

Multiply the model out: 53 a second × 3,600 seconds an hour = 190,800 an hour; × 4 hours = 763,200 a day; × 365 = 278,568,000 a year; × 4 years = 1,114,272,000, which is 1.1 × 10⁹, about a billion. In the notation the same count is 5.3 × 10¹ × 3.6 × 10³ × 4 × 3.65 × 10² × 4, and adding the exponents, 1 + 3 + 2 = 6, with the front numbers multiplying to about 1,114, gives 1.1 × 10⁹ again. A), B) and C) are each at least a power of ten too small. Remember what the model is: the four hours and the four years are guesses, and the answer is only as good as they are; but it is a billion, not a million, whichever way the guesses lean.

9 4 × 10¹⁰ locusts

Front numbers: 8 × 5 = 40. Exponents, by ADD: 7 + 2 = 9. So 40 × 10⁹, and 40 is out of range: move the point one place left, 4 × 10¹⁰, forty billion. Check: 80,000,000 × 500 = 40,000,000,000. The FAO’s count per square kilometer is real; the swarm’s area is the question’s own, and a swarm that size has been recorded.

10 2.8 × 10⁻⁴ N; close

20 × 1.4 × 10⁻⁵: the front numbers give 28, the exponent stays −5, so 28 × 10⁻⁵, and the point moves one place left, 2.8 × 10⁻⁴ N. Read back: 0.00028. The study’s measured average was 2.7 × 10⁻⁴ N, which is 0.00027, so the multiplication comes within 0.00001 N of it, about 4 percent. That is a claim tested by the arithmetic being learned: twenty hooklets at fourteen millionths of a newton each is what the measurement said it was.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Writing the notation	Move the point to just after the first nonzero digit and count the places: left is a positive exponent, right is a negative one. Read the result back to check.
3	Naming the error	ADD and SUBTRACT are for multiplying and dividing powers. Adding needs the same exponent first, then the front numbers add.
4, 5	Reading it back, and comparing	The exponent counts places to move the point. To compare, read the exponents first; the front numbers decide only when the exponents tie.
6, 7, 9	Multiplying and dividing	Front numbers multiply or divide; exponents ADD or SUBTRACT. If the front number leaves 1 to 10, move the point once and adjust the exponent by one.
8, 10	A model and a measurement	Set the calculation up in the notation and let the exponents keep count of the size. Then say what kind of number came out: a guess multiplied through, or a measurement checked.

Eight or more right: the notation is yours at both ends, and the next quiz can turn from numbers to letters. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we’ve found the right ground to work. One habit for a week: every time you write a number in the notation, read it back as an ordinary number and ask whether it is the size you meant. Then come back to these same ten.

Where this goes

Next in Creatures in Flight: Quiz 33 — Words to Symbols. A letter standing for a number you do not know yet, an expression built from words, and the one-step equation that gives the letter its value. The plumber’s flat fee from Quiz 25 becomes an expression, and the cross-multiplying promised in Quiz 24 gets its algebraic form. The pictures stay with creatures in flight.

The Company · An Interlude

One bird, both ends of the line

A ruby-throated hummingbird in flight, wings blurred, approaching a stem of yellow flowers against a soft green background — A ruby-throated hummingbird approaching a flower at the Quabbin Reservoir in Massachusetts, June 2023, in its own colors; a copy recolored into the house colors is this quiz’s masthead. Photograph by Bill Thompson, U.S. Fish and Wildlife Service. Public domain, as the work of the United States government.

This bird has no red at the throat, so it is a female or a young one; only the adult male carries the ruby. Either way it weighs about 3 grams, a little more than a penny, and it is the only hummingbird that breeds in eastern North America. In fall it flies to Central America, and many of them cross the Gulf of Mexico in a single flight, about 800 kilometers over open water in about 20 hours, having built up fat before the trip and burned most of it on the way.

Ten primary wing feathers of a female ruby-throated hummingbird laid on a blue centimeter grid, from about 3.4 centimeters long down to about 1.6 — The ten primary wing feathers of a female ruby-throated hummingbird, on a centimeter grid. From the Feather Atlas of the U.S. Fish and Wildlife Service’s National Fish and Wildlife Forensics Laboratory in Ashland, Oregon. Public domain.

The longest primary is about 3.4 centimeters, 3.4 × 10⁻² m. A feather that size is a shaft with barbs branching off it, each barb carrying rows of smaller barbules, and the barbules on one side carrying hooks that catch the barbules of the next barb over. That is what holds a feather flat against the air. In 2014 three researchers at Kiel University in Germany, Alexander Kovalev, Alexander Filippov and Stanislav Gorb, pulled the barbs of a swan’s feather apart under a force sensor and measured what the hooks were doing. A single hooklet held with about 1.4 × 10⁻⁵ newtons, and they let go in clusters of about 20 at a time, with an average force of 2.7 × 10⁻⁴ newtons; a light stroke along the feather zipped them back, which is what a bird is doing when it runs a feather through its beak. The paper is free to read in the Journal of the Royal Society Interface, and its abstract has every number used here. The Feather Atlas exists for a plainer reason: the laboratory identifies feathers that turn up in evidence, and the scans are how a feather is matched to its bird.

A hand-colored print of many ruby-throated hummingbirds hovering and perching around a trumpet creeper vine with orange and red flowers — John James Audubon, *Ruby Throated Humming Bird*, Plate 47 of *The Birds of America*, 1828, an aquatint hand-colored by Robert Havell Jr., about 27 by 40 inches. Brooklyn Museum, gift of the estate of Emily Winthrop Miles, 64.98.14; photograph by the Brooklyn Museum. Public domain.

Audubon drew the hummingbirds around a trumpet creeper, the plant the print names at its foot, with the birds in every position he had seen them take, hovering, feeding and perched, males with the ruby throat and females without. He drew every bird at life size, which is why the sheet is more than three feet tall for a bird that would fit in a spoon. A plate like this one was made by engraving the drawing onto copper, printing it, and then coloring every copy by hand, and Havell’s shop in London did that for all 435 plates of the book.

Now put the numbers of this one bird on a single line, as the Guide did. At one end, a hooklet’s grip, 1.4 × 10⁻⁵ newtons, and a barb segment, 6 × 10⁻⁵ meters. At the other, a billion wingbeats in a life, 1.1 × 10⁹, if the model’s guesses are near the truth. Fourteen powers of ten separate the ends, and the bird lives at both of them at once: every wingbeat of the billion is carried by feathers that hold together at the hundred-thousandths. Ordinary writing can hold either end, with enough zeros. It cannot hold both on one line and stay readable, and that is the whole reason a notation was invented that lets the exponent do the counting.

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